Let $f$ be your two-piece linear function. Let $\varphi\in C^\infty_0((-\epsilon,\epsilon))$ for some small $\epsilon$, such that
- $\varphi$ is even
- the integral $\int \varphi = 1$
- $x\varphi' \leq 0$
Then you can check that the convolution $\varphi*f$ is increasing, smooth, and agrees with $f$ outside $(-2\epsilon,2\epsilon)$.
Taking appropriately rescaled versions of $\varphi$ you get uniform approximations.